1
Algebra I: Strand 2. Linear Functions; Topic 13. Scavenger Hunt; Task 2.13.1
TASK 2.13.1: LOOKING FOR TREASURE
Solutions
Suppose we are given two points in a plane, A=(a,b) and B=(c,d).
1. By changing the first coordinate in point A from “a” to “t”, locate a new point C = (t, b) such
that the slope of the line through the new point and B is double that of the slope of the line
through A and B. What do you notice about the difference in the first coordinates of C and B
and the difference in the first coordinates of A and B? How does the y-intercept of the line
AB relate to the y-intercept of the line CB?
d!b
First note that the slope of the line through A and B is
. We want to find a point
c!a
d!b
C = (t, b) so that the slope of the line through C and B is
. (Note: we are looking for the
c!t
value of “t”.) Since we want to double the original slope, we get
reduces to
d!b
c!t
=2
( ) . This
d !b
c!a
( ) and so 2(c ! t) = c ! a . Thus 2c ! 2t = c ! a and !2t = !c ! a .
1
1
=2
c!t
c!a
1
c + a and C = ( 12 (c+a), b). Participants should notice that the difference
2
between the x-values of points C and B is half the difference between the x-values of points
A and B.
Therefore t =
(
)
We must now find the y-intercept of the lines AB and CB. Starting with the line AB, let I be
the y-intercept. Then
d!b
y=
x+I
c!a
Substitute the coordinates of point B for x and y.
d!b
d=
(c) + I
c!a
d!b
I=d!
(c)
c!a
Now for line CB, let I’ be the y-intercept. Then
" d ! b%
y = 2$
x+ I'
# c ! a '&
Substitute the coordinates of point B for x and y.
" d ! b%
d = 2$
(c) + I '
# c ! a '&
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University of Texas at
Austin for the Texas Higher Education Coordinating Board.
2
Algebra I: Strand 2. Linear Functions; Topic 13. Scavenger Hunt; Task 2.13.1
" d ! b%
I ' = d ! 2$
(c)
# c ! a '&
Notice that in I’, we are subtracting from d twice the amount that is being subtracted from d
in I.
2. By changing the first coordinate in point A from “a” to “q”, locate a new point D = (q, b)
such that the slope of the line through the new point and B is triple that of the line through A
and B. What do you notice about the difference in the first coordinates of D and B and the
difference in the first coordinates of A and B? How does the y-intercept of the line AB relate
to the y-intercept of the line DB?
This time we want to find the point D=(q, b) so that the slope of the line through D and B is
( )
d !b
triple that of the line through A and B. So we want d!b
. This reduces to
c!q = 3
c!a
( ) . Thus 3(c ! q) = c ! a and 3c ! 3q = c ! a . Thus !3q = !2c ! a and
1
1
=3
c!q
c!a
2
1
c + a . Therefore D = (2/3*c+1/3*a, b). Participants should note that the difference
3
3
in the x-values is one-third of the difference of the x-values of the points A and B.
q=
We must now find the y-intercept of the lines AB and DB. From before, we know that the
y-intercept of AB is
d!b
I=d!
(c)
c!a
Now for line DB, let I’ be the y-intercept. Then
" d ! b%
y = 3$
x+ I'
# c ! a '&
Substitute the coordinates of point B for x and y.
" d ! b%
d = 3$
(c) + I '
# c ! a '&
" d ! b%
I ' = d ! 3$
(c)
# c ! a '&
Notice that in I’, we are subtracting from d three times the amount that is being subtracted
from d in I.
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University of Texas at
Austin for the Texas Higher Education Coordinating Board.
3
Algebra I: Strand 2. Linear Functions; Topic 13. Scavenger Hunt; Task 2.13.1
3. By changing the first coordinate in point A from “a” to “v”, locate a new point E = (v, b)
such that the slope of the line through the new point and B is four times that of the line
through A and B. What do you notice about the difference in the first coordinates of E and B
and the difference in the first coordinates of A and B? How does the y-intercept of the line
AB relate to the y-intercept of the line EB?
This time we want to find the point E=(v, b) so that the slope of the line through E and B is
four times that of the line through A and B. So we want
( ) . This reduces to
d !b
d !b
=4
c!v
c!a
( ) . Thus 4(c ! v) = c ! a and 4c ! 4v = c ! a . Thus !4v = !3c ! a and
1
1
=4
c!v
c!a
3
1
c + a . Therefore E=(3/4*c+1/4*a, b). Participants should note that the difference in
4
4
the x-values is one-fourth of the difference of the x-values of the points A and B.
v=
We must now find the y-intercept of the lines AB and EB. From before, we know that the yintercept of AB is
d!b
I=d!
(c)
c!a
Now for line EB, let I’ be the y-intercept. Then
" d ! b%
y = 4$
x+ I'
# c ! a '&
Substitute the coordinates of point B for x and y.
" d ! b%
d = 4$
(c) + I '
# c ! a '&
" d ! b%
I ' = d ! 4$
(c)
# c ! a '&
Notice that in I’, we are subtracting from d four times the amount that is being subtracted
from d in I.
4. Let F be a point created by changing only the first coordinate of A. Make a conjecture about
the location of point F such that the slope of the line through F and B is n times the slope of
the line through A and B where n is a natural number. Explain your reasons for choosing this
point and then show that it is the correct choice.
Participants may conjecture that based on the previous exercises, the difference in the xvalues of the points F and B is 1/n times that of the points A and B. Thus the difference will
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University of Texas at
Austin for the Texas Higher Education Coordinating Board.
4
Algebra I: Strand 2. Linear Functions; Topic 13. Scavenger Hunt; Task 2.13.1
" n !1
%
1
be 1/n*(c-a). So the point will be F = $
* c + * a,b' . To show that this is true, we
n
# n
&
can calculate the slope of the line between F and B.
" d ! b%
d!b
d!b
d!b
m=
=
=
= n*$
1
1
" n !1
# c ! a '&
1 % 1
c
!
a
c
!
a
c!$
* c + * a'
n
n
n & n
# n
(
)
Thus the slope of the line between F and B is n times that of the line between A and B.
5. Make a conjecture about the y-intercept of the line FB. Explain your reasons for choosing
this value and then show that your choice was correct.
Participants should conjecture that to find the y-intercept of FB, we subtract from d, n times
the amount that is being subtracted from d in the y-intercept of AB. To show this, we must
find the y-intercept of the lines AB and DB. From before, we know that the y-intercept of
AB is
d!b
I=d!
(c)
c!a
Now for line FB, let I’ be the y-intercept. Then
" d ! b%
y = n$
x+ I'
# c ! a '&
Substitute the coordinates of point B for x and y.
" d ! b%
d = n$
(c) + I '
# c ! a '&
" d ! b%
I ' = d ! n$
(c)
# c ! a '&
Notice that in I’, we are subtracting from d, n times the amount that is being subtracted from
d in I.
6. What pattern did you notice in the y-intercepts as the slope was changed? (If you didn’t
notice any pattern, go back and make sure that you were always consistent about which point
you used to find the equation of the line. Try always using the point B.) Explain what
created the pattern. If the point B is on the y-axis, what changes?
d!b
The y-intercept of the original line is I = d !
(c) .
c!a
The y-intercept of the new line that was created by changing the slope by an integer multiple
d!b
(say k) ends up being I = d ! (k)
(c) . This is because of the fact that the y-intercept is
c!a
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University of Texas at
Austin for the Texas Higher Education Coordinating Board.
5
Algebra I: Strand 2. Linear Functions; Topic 13. Scavenger Hunt; Task 2.13.1
exactly y2 ! mx2 , where B=(x2, y2). This point is consistent throughout the process. All that
is changing is the value of m which is being multiplied by k.
If B is on the y-axis, then B is also the y-intercept of the new line. We can see that because
B = (0, d) making I, the intercept of the new line, equal to d.
Teaching notes
In this activity, it is essential that the participants always use the point B as the point used to
determine the y-intercept. Failure to do so will make it much harder to observe the relationship
that is sought.
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University of Texas at
Austin for the Texas Higher Education Coordinating Board.
6
Algebra I: Strand 2. Linear Functions; Topic 13. Scavenger Hunt; Task 2.13.1
TASK 2.13.1: LOOKING FOR TREASURE
Note: In this activity, we are specifically looking for the pattern that occurs naturally. The
“Big Idea” is what’s important here.
Suppose we are given two points in a plane, A=(a,b) and B=(c,d).
1. By changing the first coordinate in point A from “a” to “t”, locate a new point C = (t, b)
such that the slope of the line through the new point and B is double that of the slope of
the line through A and B. What do you notice about the difference in the first
coordinates of C and B and the difference in the first coordinates of A and B? How
does the y-intercept of the line AB relate to the y-intercept of the line CB?
2. By changing the first coordinate in point A from “a” to “q”, locate a new point
D = (q, b) such that the slope of the line through the new point and B is triple that of the
line through A and B. What do you notice about the difference in the first coordinates
of D and B and the difference in the first coordinates of A and B? How does the yintercept of the line AB relate to the y-intercept of the line DB?
3. By changing the first coordinate in point A from “a” to “v”, locate a new point
E = (v, b) such that the slope of the line through the new point and B is four times that
of the line through A and B. What do you notice about the difference in the first
coordinates of E and B and the difference in the first coordinates of A and B? How does
the y-intercept of the line AB relate to the y-intercept of the line EB?
4. Let F be a point created by changing only the first coordinate of A. Make a conjecture
about the location of point F given that the slope of the line through F and B is n times
the slope of the line through A and B where n is a natural number. Explain your
reasons for choosing this point and then show that it is the correct choice.
5. Make a conjecture about the y-intercept of the line FB. Explain your reasons for
choosing this value and then show that your choice was correct.
6. What pattern did you notice in the y-intercepts as the slope was changed? (If you didn’t
notice any pattern, go back and make sure that you were always consistent about which
point you used to find the equation of the line. Try always using the point B.) Explain
what created the pattern. If the point B is on the y-axis, what changes?
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University of Texas at
Austin for the Texas Higher Education Coordinating Board.